Integrand size = 28, antiderivative size = 1092 \[ \int x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right ) \, dx =\text {Too large to display} \]
-4/9*b^2*e^(3/2)*m*n^2*arctan(x*f^(1/2)/e^(1/2))*(a+b*ln(c*x^n))/f^(3/2)+1 /3*b*(-e)^(3/2)*m*n*(a+b*ln(c*x^n))^2*ln(1-x*f^(1/2)/(-e)^(1/2))/f^(3/2)-1 60/27*b^3*e*m*n^3*x/f+4/27*b^3*e^(3/2)*m*n^3*arctan(x*f^(1/2)/e^(1/2))/f^( 3/2)+2/3*b^3*(-e)^(3/2)*m*n^3*polylog(3,-x*f^(1/2)/(-e)^(1/2))/f^(3/2)-2/3 *b^3*(-e)^(3/2)*m*n^3*polylog(3,x*f^(1/2)/(-e)^(1/2))/f^(3/2)+2*b^3*(-e)^( 3/2)*m*n^3*polylog(4,-x*f^(1/2)/(-e)^(1/2))/f^(3/2)-2*b^3*(-e)^(3/2)*m*n^3 *polylog(4,x*f^(1/2)/(-e)^(1/2))/f^(3/2)+52/9*a*b^2*e*m*n^2*x/f+52/9*b^3*e *m*n^2*x*ln(c*x^n)/f-8/3*b*e*m*n*x*(a+b*ln(c*x^n))^2/f+2/9*b^2*n^2*x^3*(a+ b*ln(c*x^n))*ln(d*(f*x^2+e)^m)-1/3*b*n*x^3*(a+b*ln(c*x^n))^2*ln(d*(f*x^2+e )^m)-1/3*(-e)^(3/2)*m*(a+b*ln(c*x^n))^3*ln(1-x*f^(1/2)/(-e)^(1/2))/f^(3/2) +1/3*(-e)^(3/2)*m*(a+b*ln(c*x^n))^3*ln(1+x*f^(1/2)/(-e)^(1/2))/f^(3/2)-4/9 *b^2*m*n^2*x^3*(a+b*ln(c*x^n))+4/9*b*m*n*x^3*(a+b*ln(c*x^n))^2-1/3*b*(-e)^ (3/2)*m*n*(a+b*ln(c*x^n))^2*ln(1+x*f^(1/2)/(-e)^(1/2))/f^(3/2)-2/3*b^2*(-e )^(3/2)*m*n^2*(a+b*ln(c*x^n))*polylog(2,-x*f^(1/2)/(-e)^(1/2))/f^(3/2)+b*( -e)^(3/2)*m*n*(a+b*ln(c*x^n))^2*polylog(2,-x*f^(1/2)/(-e)^(1/2))/f^(3/2)-b *(-e)^(3/2)*m*n*(a+b*ln(c*x^n))^2*polylog(2,x*f^(1/2)/(-e)^(1/2))/f^(3/2)+ 2/9*I*b^3*e^(3/2)*m*n^3*polylog(2,-I*x*f^(1/2)/e^(1/2))/f^(3/2)+2/3*b^2*(- e)^(3/2)*m*n^2*(a+b*ln(c*x^n))*polylog(2,x*f^(1/2)/(-e)^(1/2))/f^(3/2)-2*b ^2*(-e)^(3/2)*m*n^2*(a+b*ln(c*x^n))*polylog(3,-x*f^(1/2)/(-e)^(1/2))/f^(3/ 2)+2*b^2*(-e)^(3/2)*m*n^2*(a+b*ln(c*x^n))*polylog(3,x*f^(1/2)/(-e)^(1/2...
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2544\) vs. \(2(1092)=2184\).
Time = 0.58 (sec) , antiderivative size = 2544, normalized size of antiderivative = 2.33 \[ \int x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right ) \, dx=\text {Result too large to show} \]
(54*a^3*e*Sqrt[f]*m*x - 216*a^2*b*e*Sqrt[f]*m*n*x + 468*a*b^2*e*Sqrt[f]*m* n^2*x - 480*b^3*e*Sqrt[f]*m*n^3*x - 18*a^3*f^(3/2)*m*x^3 + 36*a^2*b*f^(3/2 )*m*n*x^3 - 36*a*b^2*f^(3/2)*m*n^2*x^3 + 16*b^3*f^(3/2)*m*n^3*x^3 - 54*a^3 *e^(3/2)*m*ArcTan[(Sqrt[f]*x)/Sqrt[e]] + 54*a^2*b*e^(3/2)*m*n*ArcTan[(Sqrt [f]*x)/Sqrt[e]] - 36*a*b^2*e^(3/2)*m*n^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]] + 12* b^3*e^(3/2)*m*n^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]] + 162*a^2*b*e^(3/2)*m*n*ArcT an[(Sqrt[f]*x)/Sqrt[e]]*Log[x] - 108*a*b^2*e^(3/2)*m*n^2*ArcTan[(Sqrt[f]*x )/Sqrt[e]]*Log[x] + 36*b^3*e^(3/2)*m*n^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x ] - 162*a*b^2*e^(3/2)*m*n^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]^2 + 54*b^3* e^(3/2)*m*n^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]^2 + 54*b^3*e^(3/2)*m*n^3* ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]^3 + 162*a^2*b*e*Sqrt[f]*m*x*Log[c*x^n] - 432*a*b^2*e*Sqrt[f]*m*n*x*Log[c*x^n] + 468*b^3*e*Sqrt[f]*m*n^2*x*Log[c*x ^n] - 54*a^2*b*f^(3/2)*m*x^3*Log[c*x^n] + 72*a*b^2*f^(3/2)*m*n*x^3*Log[c*x ^n] - 36*b^3*f^(3/2)*m*n^2*x^3*Log[c*x^n] - 162*a^2*b*e^(3/2)*m*ArcTan[(Sq rt[f]*x)/Sqrt[e]]*Log[c*x^n] + 108*a*b^2*e^(3/2)*m*n*ArcTan[(Sqrt[f]*x)/Sq rt[e]]*Log[c*x^n] - 36*b^3*e^(3/2)*m*n^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c *x^n] + 324*a*b^2*e^(3/2)*m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]*Log[c*x^n ] - 108*b^3*e^(3/2)*m*n^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]*Log[c*x^n] - 162*b^3*e^(3/2)*m*n^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]^2*Log[c*x^n] + 16 2*a*b^2*e*Sqrt[f]*m*x*Log[c*x^n]^2 - 216*b^3*e*Sqrt[f]*m*n*x*Log[c*x^n]...
Time = 1.72 (sec) , antiderivative size = 1085, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2825, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right ) \, dx\) |
| \(\Big \downarrow \) 2825 |
| \(\displaystyle -2 f m \int \left (\frac {\left (a+b \log \left (c x^n\right )\right )^3 x^4}{3 \left (f x^2+e\right )}-\frac {b n \left (a+b \log \left (c x^n\right )\right )^2 x^4}{3 \left (f x^2+e\right )}+\frac {2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) x^4}{9 \left (f x^2+e\right )}-\frac {2 b^3 n^3 x^4}{27 \left (f x^2+e\right )}\right )dx+\frac {2}{9} b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )-\frac {1}{3} b n x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )-\frac {2}{27} b^3 n^3 x^3 \log \left (d \left (e+f x^2\right )^m\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2}{27} b^3 n^3 \log \left (d \left (f x^2+e\right )^m\right ) x^3+\frac {1}{3} \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (f x^2+e\right )^m\right ) x^3-\frac {1}{3} b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (f x^2+e\right )^m\right ) x^3+\frac {2}{9} b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (f x^2+e\right )^m\right ) x^3-2 f m \left (-\frac {8 n^3 x^3 b^3}{81 f}+\frac {80 e n^3 x b^3}{27 f^2}-\frac {2 e^{3/2} n^3 \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) b^3}{27 f^{5/2}}-\frac {26 e n^2 x \log \left (c x^n\right ) b^3}{9 f^2}-\frac {i e^{3/2} n^3 \operatorname {PolyLog}\left (2,-\frac {i \sqrt {f} x}{\sqrt {e}}\right ) b^3}{9 f^{5/2}}+\frac {i e^{3/2} n^3 \operatorname {PolyLog}\left (2,\frac {i \sqrt {f} x}{\sqrt {e}}\right ) b^3}{9 f^{5/2}}-\frac {(-e)^{3/2} n^3 \operatorname {PolyLog}\left (3,-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) b^3}{3 f^{5/2}}+\frac {(-e)^{3/2} n^3 \operatorname {PolyLog}\left (3,\frac {\sqrt {f} x}{\sqrt {-e}}\right ) b^3}{3 f^{5/2}}-\frac {(-e)^{3/2} n^3 \operatorname {PolyLog}\left (4,-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) b^3}{f^{5/2}}+\frac {(-e)^{3/2} n^3 \operatorname {PolyLog}\left (4,\frac {\sqrt {f} x}{\sqrt {-e}}\right ) b^3}{f^{5/2}}-\frac {26 a e n^2 x b^2}{9 f^2}+\frac {2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right ) b^2}{9 f}+\frac {2 e^{3/2} n^2 \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right ) b^2}{9 f^{5/2}}+\frac {(-e)^{3/2} n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) b^2}{3 f^{5/2}}-\frac {(-e)^{3/2} n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt {f} x}{\sqrt {-e}}\right ) b^2}{3 f^{5/2}}+\frac {(-e)^{3/2} n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) b^2}{f^{5/2}}-\frac {(-e)^{3/2} n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,\frac {\sqrt {f} x}{\sqrt {-e}}\right ) b^2}{f^{5/2}}-\frac {2 n x^3 \left (a+b \log \left (c x^n\right )\right )^2 b}{9 f}+\frac {4 e n x \left (a+b \log \left (c x^n\right )\right )^2 b}{3 f^2}-\frac {(-e)^{3/2} n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) b}{6 f^{5/2}}+\frac {(-e)^{3/2} n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (\frac {\sqrt {f} x}{\sqrt {-e}}+1\right ) b}{6 f^{5/2}}-\frac {(-e)^{3/2} n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) b}{2 f^{5/2}}+\frac {(-e)^{3/2} n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {f} x}{\sqrt {-e}}\right ) b}{2 f^{5/2}}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^3}{9 f}-\frac {e x \left (a+b \log \left (c x^n\right )\right )^3}{3 f^2}+\frac {(-e)^{3/2} \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{6 f^{5/2}}-\frac {(-e)^{3/2} \left (a+b \log \left (c x^n\right )\right )^3 \log \left (\frac {\sqrt {f} x}{\sqrt {-e}}+1\right )}{6 f^{5/2}}\right )\) |
(-2*b^3*n^3*x^3*Log[d*(e + f*x^2)^m])/27 + (2*b^2*n^2*x^3*(a + b*Log[c*x^n ])*Log[d*(e + f*x^2)^m])/9 - (b*n*x^3*(a + b*Log[c*x^n])^2*Log[d*(e + f*x^ 2)^m])/3 + (x^3*(a + b*Log[c*x^n])^3*Log[d*(e + f*x^2)^m])/3 - 2*f*m*((-26 *a*b^2*e*n^2*x)/(9*f^2) + (80*b^3*e*n^3*x)/(27*f^2) - (8*b^3*n^3*x^3)/(81* f) - (2*b^3*e^(3/2)*n^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(27*f^(5/2)) - (26*b^ 3*e*n^2*x*Log[c*x^n])/(9*f^2) + (2*b^2*n^2*x^3*(a + b*Log[c*x^n]))/(9*f) + (2*b^2*e^(3/2)*n^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*(a + b*Log[c*x^n]))/(9*f^( 5/2)) + (4*b*e*n*x*(a + b*Log[c*x^n])^2)/(3*f^2) - (2*b*n*x^3*(a + b*Log[c *x^n])^2)/(9*f) - (e*x*(a + b*Log[c*x^n])^3)/(3*f^2) + (x^3*(a + b*Log[c*x ^n])^3)/(9*f) - (b*(-e)^(3/2)*n*(a + b*Log[c*x^n])^2*Log[1 - (Sqrt[f]*x)/S qrt[-e]])/(6*f^(5/2)) + ((-e)^(3/2)*(a + b*Log[c*x^n])^3*Log[1 - (Sqrt[f]* x)/Sqrt[-e]])/(6*f^(5/2)) + (b*(-e)^(3/2)*n*(a + b*Log[c*x^n])^2*Log[1 + ( Sqrt[f]*x)/Sqrt[-e]])/(6*f^(5/2)) - ((-e)^(3/2)*(a + b*Log[c*x^n])^3*Log[1 + (Sqrt[f]*x)/Sqrt[-e]])/(6*f^(5/2)) + (b^2*(-e)^(3/2)*n^2*(a + b*Log[c*x ^n])*PolyLog[2, -((Sqrt[f]*x)/Sqrt[-e])])/(3*f^(5/2)) - (b*(-e)^(3/2)*n*(a + b*Log[c*x^n])^2*PolyLog[2, -((Sqrt[f]*x)/Sqrt[-e])])/(2*f^(5/2)) - (b^2 *(-e)^(3/2)*n^2*(a + b*Log[c*x^n])*PolyLog[2, (Sqrt[f]*x)/Sqrt[-e]])/(3*f^ (5/2)) + (b*(-e)^(3/2)*n*(a + b*Log[c*x^n])^2*PolyLog[2, (Sqrt[f]*x)/Sqrt[ -e]])/(2*f^(5/2)) - ((I/9)*b^3*e^(3/2)*n^3*PolyLog[2, ((-I)*Sqrt[f]*x)/Sqr t[e]])/f^(5/2) + ((I/9)*b^3*e^(3/2)*n^3*PolyLog[2, (I*Sqrt[f]*x)/Sqrt[e...
3.2.11.3.1 Defintions of rubi rules used
Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_. )]*(b_.))^(p_.)*((g_.)*(x_))^(q_.), x_Symbol] :> With[{u = IntHide[(g*x)^q* (a + b*Log[c*x^n])^p, x]}, Simp[Log[d*(e + f*x^m)^r] u, x] - Simp[f*m*r Int[x^(m - 1)/(e + f*x^m) u, x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m , n, q}, x] && IGtQ[p, 0] && RationalQ[m] && RationalQ[q]
\[\int x^{2} {\left (a +b \ln \left (c \,x^{n}\right )\right )}^{3} \ln \left (d \left (f \,x^{2}+e \right )^{m}\right )d x\]
\[ \int x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x^{2} \log \left ({\left (f x^{2} + e\right )}^{m} d\right ) \,d x } \]
integral((b^3*x^2*log(c*x^n)^3 + 3*a*b^2*x^2*log(c*x^n)^2 + 3*a^2*b*x^2*lo g(c*x^n) + a^3*x^2)*log((f*x^2 + e)^m*d), x)
Timed out. \[ \int x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right ) \, dx=\text {Timed out} \]
Exception generated. \[ \int x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right ) \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x^{2} \log \left ({\left (f x^{2} + e\right )}^{m} d\right ) \,d x } \]
Timed out. \[ \int x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right ) \, dx=\int x^2\,\ln \left (d\,{\left (f\,x^2+e\right )}^m\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3 \,d x \]